LPC Physics
Updated 09/06
Hooke’s Law and Simple Harmonic Motion
Hooke’s Law and Simple Harmonic Motion
Purpose:
In this lab you will explore the behavior of an oscillating spring and mass system. Along
the way, you will learn how the system deviates from an ideal system by considering the
effects of the spring mass and damping.
Specifically, you will
 measure the spring constant for a spring
 determine the period of oscillation of the system
 determine the spring constant while the spring is in motion
 verify theory of a damped oscillator and determine the damping constant
Equipment:
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Pendulum Hanger
Hooke’s Law Spring
Hooked Mass Set
LabPro Kit
Motion Detector
Force Probe
Clamps and Rods
Meter Stick
Balance
Theory:
Hooke’s Law and Springs:
A Hooke's Law Spring is defined by the relation
F   kx
Eq. 1
where x is the displacement of the spring from its equilibrium position. From Newton's
2nd Law,
d 2x
F  ma  m 2  kx
dt
for a spring attached to a mass. The solution to this equation is
x  x max sin  o t   
where  o  mk and  is a phase constant, determined from initial conditions.
A mass hanging from a massless spring oscillates about its equilibrium position with a
period, T, given by
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T=
2
0
m
k
 2
Eq. 2
However, if the spring is not massless, then m must be replaced with m + amsp where a
equals some fraction of the spring mass. Thus, in general, the period of a spring/mass
system can by described by
T  2
m  amsp
Eq. 3
k
Eq. 3 can be solved for m, so that
m
T2
k  am sp
4 2
This is an equation of the form y = mx + yo, where x =
Eq. 4
T2
4 2
, and yo = amsp .
Note that the quantity (m + amsp) is known as the “equivalent mass” of the system. For
an “ideal” Hooke’s law spring, a = 1/3.
According the an article in the Physics Teacher (April 2000) by Nathaniel R. Greene, and
Ryan J. Dunn of Bloomsberg Univeristy, Pa, a conical spring hung "wide end up" has a
theoretical value for a of .401 vs. a theoretical value of a = .276 for the "thin end up."
Damped oscillations
If the spring experiences a damping force such as air resistance that varies linearly with
speed, such that:
F  bv
where b is the damping coefficient, and v is the speed of the mass (v = - dx/dt), in which
case:
m
dx
d 2x
  kx  b
2
dt
dt
Eq. 5
The solution to which is:
b
t
x  x max e 2 m cos(t   )
where....
  02  (
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b 2
)
2m
Eq. 6
LPC Physics
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Hooke’s Law and Simple Harmonic Motion
The phase constant and xmax are determined by initial conditions. A graph of a similar
function to Eq. 5 is shown below:
Note that and two successive amplitudes are related by:
x(t 2 )
 b (t )
 e 2m
x(t1 )
Eq. 7
where T = t2 - t1 = one full period.
Experiment:
Part A: Determination of the Spring Constant
1. Set up rods and clamps so the spring will hang freely, as shown in Figure 1 below.
2. Determine the mass of the spring using a digital scale.
3. Starting at 100 grams, add 5 weights in 100g increments up to about 600g and
determine how much each 100 g mass stretches the spring. (You can attach a meter
stick to the apparatus to make the measurements). Record your values in Table 1.
mass on spring
Figure 1 Basic Experimental Set-Up
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Part B: Determination of the Period of Oscillation
1. Connect the AC adapter to the LabPro by inserting the round plug on the 6-volt
power supply into the side of the interface. Shortly after plugging the power supply
into the outlet, the interface will run through a self-test. You will hear a series of
beeps and blinking lights (red, yellow, then green) indicating a successful startup.
2. Attach the LabPro to the computer using the USB cable that is Velcro-ed to the side
of the computer box (do not unplug the USB cable from the computer!). The LabPro
computer connection is located on the right side of the interface. Slide the door on
the computer connection to the right and plug the square end of the USB cable into
the LabPro USB connection.
3. Connect a motion detector to a digital port (DIG/SONIC) on the LabPro. The digital
ports, which accept British Telecom-style plugs with a left-hand connector, are
located on the same side as the computer connections. If you are using an older
motion detector, you may need to remove the flat gray cable from the sensor, and
replace it with the round black cable labeled Motion Detector Cable, with British
Telecom plugs on both ends. Place the motion detector on the floor below the
hanging mass, keeping in mind that the detector will not register any motion that
occurs closer than 0.4 m from its sensor.
4. Start the Logger Pro program. You should see a screen that displays graphs for
distance vs. time, velocity vs. time and acceleration vs. time. You can ignore the
acceleration vs. time graph for the first part of this experiment.
mass on spring
motion detector
Figure 2 Basic Experimental Set-Up with Motion Detector
5. Place a 100 to 300 gram mass on the spring and set it oscillating. Verify that the
motion detector is working by hitting Collect. Experiment with the setup until you
see a sine curve on the display. Try adjusting the Experiment > Data Collection
setting to produce the smoothest curve.
6. Take data for each of the three masses you used in Part A. Determine the period of
oscillation for each mass by highlighting a region of the graph from one oscillation
peak to another oscillation peak. The time interval will appear in the bottom lefthand corner of the graph window as “dx: ”. For best accuracy, determine the time
for as many complete oscillations as possible, and divide by the number of
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oscillations to determine the period. Record your data in Table 2 (once again
neglecting the spring mass).
7. Save graphs of your data in the Physics 8A folder on the desktop.
Extra Credit (1 pt)
Show theoretically that a = 1 .
3
Hint: Assume that all portions of the spring oscillate in phase and that the velocity of a
segment dx is proportional to the distance from the fixed end, that is:
x
vx  v .
l
m
Also, note that the mass of a segment of the spring is dm  dx . Compare your result
l
1
2
with K  mv for a massless spring (i.e., m should be replaced by m + amsp.)
2
Part C: Verifying Hooke’s Law and the Experimental Value of k
1. Replace the pendulum hanger with a rod, and attach the force probe, as shown in
Figure 3 below. You will probably need to use the DIN-BTA adapter in the LabPro
Kit to connect the force probe to the LabPro.
student force probe
mass on spring
motion detector
Figure 3 Experimental Set-Up with Motion Detector and Student Force Probe
2. Connect the force probe to Channel 1 of the LabPro. The motion detector should
remain attached to DIG/SONIC1.
3. If you do not have an auto-ID sensor (which is the likely case), a dialog box will pop
up asking you to confirm the sensors being used. If you have the suggested sensors
attached to the LabPro in the suggested ports, click “OK”. If the “OK” button is not
active, ask your instructor for help.
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4. Calibrate the Force probe by choosing Experiment > Set Up Sensors > LabPro 1.
Then click on the Student Force sensor icon, and choose Calibrate. Make sure the
“Current Calibration” is chosen as “Student Force N <Computer>”, and that “Live
Calibration” is chosen in the highlighted drop-down menu. The box marked “One
Point Calibration” should not be checked. When you are ready, click Calibrate
Now. One value may be 0N (easy!). You will need to add another known force by
hanging a hooked mass from the sensor…be sure to convert from grams to Newtons!
5. Attach the spring to the force probe and set a small mass oscillating. Experiment with
the setup, and adjust the sampling rate to get the smoothest curves possible.
6. When you are satisfied with your setup, run the experiment. When it is finished
gathering data, select the data table and copy and paste the data into Graphical
Analysis or Excel.
7. Repeat Steps 5-6 for the same masses as in Parts A and B. Save your graphs in the
Physics 8A folder on the desktop, and complete Table 4.
Part D: The Damped Oscillator
1. Remove the force probe and configure the spring, mass and motion detector as in Part
B.
2. Place a 50 to 100 gram mass on the spring and experiment with the system until a
smooth sine curve for displacement vs. time is visible on the display. In this case, you
will need to consider the trade-off between data rate and number of cycles. The
higher the data rate, the more accurate your measurements, and smaller number of
cycles that can be measured. For this experiment a larger number of cycles is desired.
Another recommended approach is to use a stopwatch in conjunction with the
computer, i.e. start the program and the stopwatch simultaneously, when the program
stops, save the graph and then start the program again, making sure to take down the
reading from the stopwatch when you start the program. Do two trials using the same
mass, waiting 30 seconds to a minute, and starting the program again, making sure to
keep a "running" track of time. Do this over several minutes and you will get several
curves that better show the damping effects (i.e. values of x1 and x2 and corresponding
values of t1 and t2 that are separated by a sufficiently long time interval to reveal the
effects of damping.)
3. Run the experiment. When the detector shuts off, select Analyze > Examine to
measure points from the graph. Measure the time, and displacement for each peak
(amplitude) on your graph. Record your values for t1, t2, x1 and x2 in Table 5. A few
mass values have been entered as an example. You can use any two times for t1 and
t2, but note that the value of T in Eq. 7 should be replaced with t2 - t1. You may want
to adjust the data rate to give the best values --i.e. so that at the very least, the
amplitude values are decreasing.
Part E: Interactive Physics
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1. Open up Interactive Physics > Physics Investigations > Oscillations > Damped
Harmonic Motion.
2. You should see a circle connected to a spring and a “damper.” Click Run and
observe it for a few cycles. Hit Stop when you are done. Click on the graph and
stretch it out so that about six or seven seconds of time are visible.
3. Reset. Double click on the mass to view its properties. Record its mass. Double
Click on the spring and record its spring constant. Double Click on the Damper and
record its damping constant. When you are through, Run the simulation. Stop
before it reaches the end of the time axis. Select View > Workspace and check
Rulers and coordinates. Make are least four separate measurements of t1 and t2 as in
Part D above, and determine the damping constant b using the two different methods
used earlier (i.e. using Eqs 6 and 7).
Enter your results:
Average value of damping constant from
period measurement (Eq. 6):
Average value of b from amplitude
measurement (Eq. 7):
4. Describe your results in your lab report.
5. Now select the damper and delete it. Run the simulation again and measure the
period. Calculate the angular frequency. Does it agree with the predicted value?
6. Now setup your own oscillator experiment using interactive physics. Your instructor
will help you out if you run into difficulties. Describe it and its results.
Analysis:
Part A
1. Use Graphical Analysis or Excel to plot weight (mg) vs. displacement of the spring.
Note that the starting point isn’t important. Determine the slope of your line using the
curve-fitting tool. The slope of your graph is the spring constant k (why?).
Enter your value of k = ___________.
Don’t worry about uncertainty in these values for now.
2. Save a copy of your graph in the Physics 8A folder on the computer desktop.
Part B
1. For 500 grams, your period and spring constant most likely will not agree with the
values you determined earlier. This is most likely due to the spring’s (unaccounted)
mass. To correct this problem, use your values of T, and m from Table 2 to complete
Table 3. Then determine the correct value of the constant a by graphing Eq. 4 using
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the data in Table 3. Graph your equation and determine the y intercept. In this case,
2
your “y” values are the masses, the “x” values are the quantity 4T 2 and the y intercept
of your graph is: am sp . Dividing the y intercept by the spring mass will give the best
value of a.
2. Compare your value of a to the theoretical value of a = 1/3. Are they equal to within
your ability to measure?
Part C
1. Graph force vs. distance and fit a straight line curve to the results. The slope of this
curve should be your spring constant. (Why?)
2. Does this value of k agree with the value from Part A within your uncertainty? If not,
why not?
3. Print out your graphs, and turn them in with your write up.
Part D
1. Rewrite Eq. 7 to solve for the damping constant b. Plug in appropriate values from
Table 5 to solve for the damping constant. Enter your results in the data table.
2. Using your values of time, determine the period of your oscillator, and the angular
frequency. Use Eq. 6 to calculate the damping constant b. If you think that the spring
mass should be used in the calculation, be sure to use the effective mass in Eq. 6.
3. Do your values of b agree from the two different methods (i.e. Steps 4 and 5 above)?
If they are close, use your data to determine the uncertainty in the measurements and
compare the values within the experimental uncertainty. If they aren’t close, suggest
an explanation why. Show your calculations and describe your results in your lab
report.
Results:
Write at least one paragraph describing the following:
 what you expected to learn about the lab (i.e. what was the reason for conducting
the experiment?)
 your results, and what you learned from them
 Think of at least one other experiment might you perform to verify these results
 Think of at least one new question or problem that could be answered with the
physics you have learned in this laboratory, or be extrapolated from the ideas in
this laboratory.
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Clean-Up:
Before you can leave the classroom, you must clean up your equipment, and have your
instructor sign below. How you divide clean-up duties between lab members is up to you.
Clean-up involves:
 Completely dismantling the experimental setup
 Removing tape from anything you put tape on
 Drying-off any wet equipment
 Putting away equipment in proper boxes (if applicable)
 Returning equipment to proper cabinets, or to the cart at the front of the room
 Throwing away pieces of string, paper, and other detritus (i.e. your water bottles)
 Shutting down the computer
 Anything else that needs to be done to return the room to its pristine, pre lab form.
I certify that the equipment used by ________________________ has been cleaned up.
(student’s name)
______________________________ , _______________.
(instructor’s name)
(date)
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Data Tables:
TABLE 1
mass (m)
kg
weight (mg) Newtons
distance (x)
meters
TABLE 2
mass (m)
kg
period of oscillation (T)
seconds (from graph)
spring constant (k)
N/m (from calculation)
TABLE 3
y = Mass (m)
kg
Enter your results:
Period (T)
seconds
x=
T2
2
4
y intercept = ______________
a = ______________
TABLE 4
Mass
Enter your results:
Spring Constant
(from graph)
Average Value of k = ______________
Uncertainty in k = ______________
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(deviation from the average)
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Hooke’s Law and Simple Harmonic Motion
TABLE 5
mass (m)
kg
time 1
time 2
X1
X2
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b (from
eq VI)
b (from
eq V)
difference